Question Number 151247 by mathdanisur last updated on 19/Aug/21
$$\frac{\mathrm{1}}{\mathrm{5}}\:+\:\frac{\mathrm{1}}{\mathrm{5}^{\mathrm{2}} \centerdot\mathrm{2}}\:+\:\frac{\mathrm{1}}{\mathrm{5}^{\mathrm{3}} \centerdot\mathrm{3}}\:+\:\frac{\mathrm{1}}{\mathrm{5}^{\mathrm{4}} \centerdot\mathrm{4}}\:+\:…\:=\:? \\ $$
Answered by qaz last updated on 19/Aug/21
$$\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{n5}^{\mathrm{n}} }=−\mathrm{ln}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{5}}\right)=\mathrm{ln}\frac{\mathrm{5}}{\mathrm{4}} \\ $$
Commented by mathdanisur last updated on 19/Aug/21
$$\mathrm{Thankyou}\:\mathrm{Ser},\:\mathrm{but}\:\mathrm{ln}\frac{\mathrm{5}}{\mathrm{4}}\:\mathrm{or}\:\mathrm{ln}\frac{\mathrm{4}}{\mathrm{5}}.? \\ $$
Commented by naka3546 last updated on 19/Aug/21
$$\mathrm{ln}\:\frac{\mathrm{5}}{\mathrm{4}} \\ $$